My model is magenta in Texturing Paint. What does it mean?
That magenta means that the material of your cat has a non assigned texture; the colored checkboard means that an UV map has been created referring to that image. 

Open a UV Image editor window, select the colored checkboard, save it to your hard drive with a name like "Cat skin";
Select your material and load the saved "cat skin" as an image texture.
Now you should be able to paint it in texture paint mode.
It should work both in Cycles and Blender Internal.

To make your cat furry you will have to add to the skin material a new "hair" particle system and play with the adjustements.

Good luck!